Quick questions on Differentiation techniques explained: H2 Mathematics Calculus

Real expressions mix the rules: a quotient whose parts are themselves products, or a chain inside a product. Identify the outermost structure first, then work inward, applying the chain rule wherever a composite appears.

The hardest H2 differentiation questions nest one rule inside another, and the reliable approach is to name the outermost operation first. For $y = x^2 \tan(3x)$, the outermost structure is a product ($u = x^2$, $v = \tan 3x$), but differentiating $v$ needs the chain rule: $v' = 3\sec^2(3x)$. So $\tfrac{dy}{dx} = 2x\tan(3x) + x^2 \cdot 3\sec^2(3x)$. The discipline of asking "is the whole thing a product, a quotient, or a composite?"

The numerator is $u'v - uv'$ (top derivative first); reversing the subtraction flips the sign.

Differentiate $y = (3x + 1)^5$. [2 marks]

Differentiate $y = x \cos x$. [2 marks]

Differentiate $y = \dfrac{e^x}{x}$. [2 marks]